Global regularity for water waves in two dimensions

We will start by describing some general features of quasilinear
dispersive and wave equations. In particular we will discuss a few
important aspects related to the question of global regularity for such
equations.
We will then consider the water waves system for the evolution of a
perfect fluid with a free boundary. In 2 spatial dimensions, under the
influence of gravity, we prove the existence of global irrotational
solutions for suitably small and regular initial data. We also prove
that the asymptotic behavior of solutions as time goes to infinity is
different from linear, unlike the 3 dimensional case.